Klein bottle
The idea of a one-sided surface seems far-fetched. Nevertheless, a famous one
was discovered by the German mathematician and astronomer August Möbius in
the 19th century. The way to construct such a surface is to take a strip of paper,
give it one twist and then stick the ends together. The result is a ‘Möbius strip’, a
one-sided surface with a boundary curve. You can take your pencil and start
drawing a line along its middle. Before long you are back where you started!
It is even possible to have a one-sided surface that does not have a boundary
curve. This is the ‘Klein bottle’ named after the German mathematician Felix
Klein. What’s particularly impressive about this bottle is that it does not intersect
itself. However, it is not possible to make a model of the Klein bottle in three-
dimensional space without a physical intersection, for it properly lives in four
dimensions where it would have no intersections.
Both these surfaces are examples of what topologists call ‘manifolds’ –
geometrical surfaces that look like pieces of two-dimensional paper when small
portions are viewed by themselves. Since the Klein bottle has no boundary it is
called a ‘closed’ 2-manifold.
The Poincaré conjecture
For more than a century, an outstanding problem in topology was the
celebrated Poincaré conjecture, named after Henri Poincaré. The conjecture
centres on the connection between algebra and topology.
The part of the conjecture that remained unsolved until recently applied to
closed 3-manifolds. These can be complicated – imagine a Klein bottle with an